[Retracted] Neutrosophic Soft Quad Structures Concerning Neutrosophic Soft Points

Mehmood, Arif; Afzal, Farkhanda; Khan, Muhammad Imran; Abdullah, Saleem; Gul, Saeed

doi:https://doi.org/10.1155/2022/8483632

Mathematical Problems in Engineering

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Volume 2022 | Article ID 8483632 | https://doi.org/10.1155/2022/8483632

[Retracted] Neutrosophic Soft Quad Structures Concerning Neutrosophic Soft Points

Arif Mehmood,¹Farkhanda Afzal,²Muhammad Imran Khan,³Saleem Abdullah,⁴and Saeed Gul⁵

Academic Editor: Ewa Rak

Received02 Oct 2021

Revised02 Dec 2021

Accepted07 Dec 2021

Published19 Jan 2022

Abstract

In this article, a new concept of neutrosophic soft quad-topological structures is introduced. Generalised neutrosophic soft open sets in neutrosophic soft quad-topological structures concerning soft points of the space are introduced. Different results are addressed in neutrosophic soft quad-topological structure on the basis of these new neutrosophic soft open sets. Neutrosophic soft separation axioms and other separation axioms are addressed in neutrosophic soft quad-topological structures. The engagement of these axioms is switched over with different results with respect to soft points. Neutrosophic soft topological properties of some results are also addressed in neutrosophic soft quad-topological structure. To secure the results, examples are constructed. The nonvalidity of some results are justified with examples. The understanding of some complicated problems is secured with simple techniques.

1. Introduction

Zadeh [1] introduced fuzzy set theory. It addresses vagueness and incomplete data used in various fields of science. Atanassov [2] addressed the short comings in fuzzy set theory with cultured way and opened a door to a new idea with new title intuitionistic fuzzy set theory. Molodtsov [3] inaugurated the concept of soft set theory to address the uncertainty. Soft set theory hugged with many applications in many fields, like smoothness of function, Riemann integration, measurement theory, and game theory. Molodtsov [4] continuously work on soft set theory. Alshehri et al. [5] applied the concept of soft set theory to K-algebra and traced some characteristics of Abelian soft K-algebras. In addition, they introduced the notion of intersection K-algebras and addressed some of their properties. Maji et al. [6] initiated first practical application of soft sets in decision-making problems. Feng et al. [7] for the first time considered the combination of soft sets, fuzzy sets, and rough sets. Using soft sets as the granulation structures, Feng et al. [8] defined soft approximation spaces, soft rough approximations, and soft rough sets, which are generalizations of Pawlak’s rough set model based on soft sets. It has been proved that, in some cases, Feng’s soft rough set model could provide better approximations than classical rough sets.

Xu et al. [9] unveiled the concept of vague soft set theory which is an extension to soft set theory. Huang et al. [10] deeply studied [9] and pointed out some incorrect results. They verified the incorrect results with examples and gave some more new definitions. Chang Wang and Yaya Li [11] initiated the concept of vague soft topological structures with title topological structure of vague soft sets. The authors discussed the basic concepts related to vague soft topological space and studied the results in vague soft topology. The soft set to the hyper-soft set was generalized by Smarandache [12]. In addition to this, the author introduced the hybrids of crisp, fuzzy, intuitionistic fuzzy, neutrosophic, and plithogenic hyper-soft set. Bera and Mahapatra [13] introduced the concept of neutrosophic soft topology and discussed some fundamental results. Ozturk in [14] introduced new concepts in neutrosophic soft topological spaces. These concepts are boundary, dense set, and neutrosophic soft basis. In addition, the concept of soft subspace on neutrosophic soft topological spaces. Some interesting results are addressed with respect to soft points. Some complicated results are secured with best examples.

Yolcu et al. redefined neutrosophic soft mapping and studied the images and inverse images of neutrosophic soft sets. The authors continued to trace the basic operations and other related properties of neutrosophic soft mapping. The authors beautifully applied neutrosophic soft mapping to application in decision-making problems.

Ozturk et al. [14] are pioneers of new operations on neutrosophic soft sets. Ozturk et al. [15] unveiled the concept of neutrosophic soft mapping, neutrosophic soft open mapping, and neutrosophic soft homeomorphism on the basis of operation defined in [14]. Some results are secured with best understandable examples. Gunduz et al. [16] introduced new concepts of neutrosophic soft sets. They defined new separation axioms in neutrosophic soft topological spaces with respect to soft points. In continuation, the relationship among these neutrosophic soft axioms has been addressed. Interior and closer of neutrosophic soft sets are also addressed. On the basis of these concepts, some other structures are also discussed. Most of the difficult results are secured with best examples.

AL-Nafee [17] introduced new family of neutrosophic soft sets. The author defined new operations on the neutrosophic set. These operations are union and intersection. On the basis of these new operations, the author defined neutrosophic soft topological space. AL-Nafee et al. [18] continued their work and extended the neutrosophic soft topological space to neutrosophic soft bitopological space on the basis of operations defined in [18]. The authors regenerated all the fundamental results of NSBTS on these basic operations. Dadas and Demiralp [19] inaugurated NSBTS on the basis of the operations defined in [13]. The authors introduced pairwise neutrosophic soft (closed) sets in NSBTS. These references [13–19] became source of motivation for my new research.

In our study, we have worked with the operations given in references [14–16] which are entirely different from references [13, 17]. In Section 2, some basic recipes are inaugurated. Originality begins in Section 3. In this section, new concepts of neutrosophic soft quad-topological spaces are addressed with examples. Some results, union and intersection are also studied in neutrosophic soft quad-topological spaces. In Section 4, some important definitions of generalized neutrosophic soft open sets in neutrosophic soft quad topological spaces are introduced. These definitions are semiopen, preopen, and open sets, respectively. These definitions became source of generation of different neutrosophical soft separation axioms and neutrosophical soft other separation axioms in neutrosophic soft quad topological spaces with respect to soft points of the second space. Neutrosophical soft quad homeomorphism which is a safe carriage for different structure from one space to another is defined. Soft closer attachment with neutrosophical soft separation axioms and neutrosophical soft other separation axioms in neutrosophic soft quad topological spaces with respect to soft points of the second space are addressed. In Section 5, more main results are addressed. Among neutrosophical soft separation axioms, Hausdorff space is considered to be the most important separation axioms. Important things should be given serious attention. So, this section has almost been engaged with the study of Hausdorff space. Through neutrosophical soft function, the characteristics of one space can be migrated to another space if the neutrosophical soft function is satisfying conditions of neutrosophical soft bijections and bicontinuousness. A soft function satisfying these conditions is known as neutrosophical soft homeomorphism. Neutrosophical soft homeomorphism is giving birth to neutrosophical soft topological property. Some neutrosophical soft topological properties of Hausdorff space are addressed with respect to soft points. Sequentially, compactness and countably compact in neutrosophic soft quad topological spaces with respect to soft points of the second space are addressed. Engagement of Hausdorff space with closed sets in neutrosophic soft quad topological spaces is addressed. In Section 6, some concluding remarks and future work are given.

2. Fundamental Concepts

In this section, fundamental concepts are addressed. These fundamental concepts are neutrosophic set, soft set, neutrosophic soft set, neutrosophic soft complement, neutrosophic soft subsets, neutrosophic soft union, neutrosophic soft intersection, null neutrosophic soft, neutrosophic soft absolute set, neutrosophic soft points, and neutrosophic soft topological spaces.

Definition 1. (see [20]). A neutrosophic set (NS) symbolized by A on the key set is defined as: , where

Definition 2. (see [3]). Let be the key set, be a set of parameters, and symbolizes the power set of . A pair is referred to as a soft set (SS) over where is a map given by .

Definition 3. (see [21, 22]). Let be the key set and set of parameters. Let signifies power set of all neutrosophic sets on . Then, a neutrosophic soft set over is a set defined by a set valued function representing a mapping , where is called approximate function of the neutrosophic soft set It can be written as a set of ordered pairs: .

Definition 4. (see [13]). Let be a neutrosophic soft set over key set the complement of is signified and is defined as follows:
It is clear that

Definition 5. (see [23, 24]). Let and are two neutrosophic soft set over key set . Then, is said to be neutrosophic soft subset of if , , , , and It is denoted by Neutrosophic soft is said to be neutrosophic soft equal to if is neutrosophic soft subset of and neutrosophic soft subset of It is dented by

Definition 6. (see [16]). Let be two neutrosophic soft subsets over key set so that , then their union is denoted by and is defined as where

Definition 7. (see [16]). Let and be two neutrosophic soft subsets over key set such that , then their intersection is defined as is given as where

Definition 8. (see [14]). Let be a neutrosophic soft set over a key set , then it is said to be a null neutrosophic soft set if
It is signified as

Definition 9. (see [14]). Let be a neutrosophic soft set over a key set , then it is said to be an absolute soft set if and for all
It is signified as ; clearly, .

Definition 10. (see [14]). Let NSS be the family of all neutrosophic soft sets and , then is said to be a neutrosophic soft topology on if : the union of any number of neutrosophic soft sets in : the intersection of a finite number of neutrosophic soft sets in in ; then, ) is said to be an over

Definition 11. (see [14]). Let be the family of all neutrosophic sets over , then neutrosophic set is neutrosophic point, for and is defined as follows:

Definition 12. (see [14]). Let be the family of all neutrosophic soft sets over KS Then, neutrosophic soft set is called a neutrosophic soft point if for every and is defined as follows:

Definition 13. (see [14]). Let (be a neutrosophic soft sets over KS , then ) if

Definition 14. (see [14]). Let be a soft neutrosophic topological space over Let be a neutrosophic soft set. Then, is called a neutrosophic soft neighborhood of the neutrosophic soft point if there exists a neutrosophic soft open set such that

3. Characterisation of Neutrosophic Soft Quad-Topological Structures

In this section, the concept of neutrosophic soft quad topological space is defined. Furthermore, new types of open and closed sets have been introduced in neutrosophic soft quad topological spaces.

3.1. Definition

If are four neutrosophic soft topological space (NSTS), then is called neutrosophic soft quad topological space. A neutrosophic soft subset is said to be neutrosophic soft quad open in if there exists a neutrosophic soft open set in neutrosophic soft open set in , neutrosophic soft open set in and neutrosophic soft open in such that

3.2. Example

Let , and , , , and where , and being as follows:

Therefore, , are on and so is a NSQTS.

3.3. Theorem

Let be a NSQTS. Then, is a on

Proof. For this, we have to verify all the three conditions of neutrosophic soft quad topological space. Conditions (1) and (3) are obvious; for condition (2), let , then , , , and , , and are NSTS on , then , , and . Therefore, .

Remark 1. Let be a NSQTS, then need not be a NSQTS on

3.4. Example

Let , and , , , and , where , and being NSSs as follows:

Here, is not a on.

3.5. Definition

Let be a NSQTS. Then, an is called as a pairwise NSQ open set if there exists a NSOS in in in and in such that for all such that

3.6. Definition

Let be a NSQTS. Then, an NSSis called as a PNSOS if there exist a NSOS in and a in in , and in such that, for all ,

The set of all pairwise neutrosophic soft quad open sets in is denoted by PNSO .

3.7. Definition

Let be a NSQTS. Then, a NSSis called as a pairwise neutrosophic soft quad closed set (PNSC) if is a It is clear that is a PNSC set if there exists a in , NSOC in , in , and in such that, for all ,

The set of all in is denoted by.

3.8. Theorem

Let be a . In this case,(1)(2)If , then (3)If then

Proof. .(1)Since and , then and are (2)Since there exist , , , and such that for all Then,As are on Therefore, (3)Since there exist , and such that , for all Then,Then, as , , , and .

3.9. Definition

Let be a . The PNS closure of denoted by is the intersection of all containing i.e.,

It is clear that is the smallest containing .

3.10. Theorem

Let be a and . Then, (1) and (2) (3) is a PNSCS if (4) if (5) (6) i.e., is a PNSCS

Proof. It is obvious.

3.11. Theorem

Let be a . Then, if and only if for all , where any is contains and is the family of all contains .

Proof. Let and suppose that there exists such that . Then, . Thus, which implies , a contradiction.
Conversely, that then Therefore, by hypothesis , a contradiction.

4. Exhabitation of Some Definitions and Main Results

In this section, some important definitions of generalized neutrosophic soft open sets in neutrosophic soft quad topological spaces are introduced which pave the way to our new results. These definitions are semiopen, preopen, and open sets, respectively. These definitions became source of generation of different neutrosophical soft separation axioms and neutrosophical soft other separation axioms in neutrosophic soft quad topological spaces with respect to soft points of the second space. Neutrosophical soft quad homeomorphism which is a safe carriage for different structure from one space to another is defined. Soft closer attachment with neutrosophical soft separation axioms and neutrosophical soft other separation axioms in neutrosophic soft quad topological spaces with respect to soft points of the second space are addressed.

Definition 15. Let be a over be a set over Then, is(1)Neutrosophic soft quad semiopen if (2): Neutrosophic soft quad preopen if (3): Neutrosophic soft quad open if and neutrosophic soft quad close if

Definition 16. Let be a over , are points if there exist -open sets and such that
Then, is called a space.

Definition 17. Let be a over , are points. If there exist -open sets such that
then is called a space.

Definition 18. Let over are points. If there exists open set such that , and then is called a space.

Definition 19. Let and be two A NS function : is said to homeomorphism if (i) is bijective; (ii) is continuous; (iii) is continuous or is NSO or is NSC.

Theorem 1. Let be a over the father set , then is Q space if and only if for distinct points and , there exists an -open set containing but not such that

Proof. Let be two points in space, then there exists disjoint open sets such that and since and Next suppose that, then there exists a open set containing but not such that that is and are mutually exclusive open sets supposing and in turn.

Theorem 2. Let be a Then, is space if every point if there exists an open set such that then an space.

Proof. Suppose . Since is space. and are closed sets in ; then, Thus, there exists a open set such that So, we have and that is, is a space.

Definition 20. Let be a . be a closed set and . If there exists open sets and such that and , then is called a -regular space. is said to be space, if is both a NSQ regular and space.

Theorem 3. Let be a is space iff for every , that is, ( such that

Proof. Let is space and Since is space for the NSQ point and closed set , there exists and such that and . Then, we have Since closed set. Conversely, let and be a closed set. and we have . Thus, and So, is space.

5. Characterization of More Results Concerning New Definition

In this section, more main results are addressed. Among neutrosophical soft quad separation axioms, Hausdorff space is considered to be the most important separation axioms. Important things should be given serious attention. So, this section has almost been engaged with the study of Hausdorff space. Through neutrosophical soft quad function, the characteristics of one space can be migrated to another space if the neutrosophical soft quad function is satisfying conditions of neutrosophical soft quad bijections and bicontinuousness. A soft quad function satisfying these conditions is known as neutrosophical soft quad homeomorphism. Neutrosophical soft quad homeomorphism is giving birth to neutrosophical soft quad topological property. Some neutrosophical soft quad topological properties of Hausdorff space are addressed with respect to soft points. Sequentially compactness and countably compact in neutrosophic soft quad topological spaces with respect to soft points of the second space are addressed. Engagement of Hausdorff space with closed sets in neutrosophic soft quad topological spaces is addressed.

Theorem 4. Let be such that it is Hausdorff space and be another such that let : be homeomorphism. Then, is also of the same behavior of Hausdorffness.

Proof. Let such that . Since : be NSQ homeomorphism. So, there exists such that . Since is Hausdorff space, so there exist and in Hausdorff space such that and such that Hence, Since is NSQ homeomorphism, so is NSQ open. Also, This proves is also of characteristics of Hausdorffness.

Theorem 5. Let be and be another which satisfies one more condition of Hausdorffness. Let : a function such that it is homeomorphism, then is also of characteristics of Hausdorffness.

Proof. Suppose such that . Since NSQ homeomorphism, so Let such that .
Let
Let such that . Since is Hausdorff space. There definitely exists open sets and such that and with .
are open in Now, So is is Hausdorff space.

Theorem 6. Let be and be another . Let : be a mapping. Let is Hausdorff space, then it is guaranteed that is a closed subset of .

Proof. Given that be and be another . Let be a mapping such that it is continuous mapping is Hausdorff space Then, we will prove that is NSQ close subset of Equivalently, we will prove that is an open subset of . Let Then, or accordingly. Since, is NSQ Hausdorff space. Certainly, are NS points of , so there exists open sets such that , provided . Since, is soft continuous, are open sets in So, is open in Now, we show that For this, let and and . Since Thus, is open in So, this implies that is interior point of . So, every point of is interior point of . So is open in

Theorem 7. Let be and be another . Let : be an open mapping such that it is onto. If the soft set is closed in then is Hausdorff space.

Proof. Suppose be two points of such or Then, , that is, . Since open in there exist open sets and in such that . Then, since is open, are open in containing , and as disjoint. It follows that is Hausdorff space.

Theorem 8. Let be a second countable space and let be uncountable subset of , then at least one point of is a limit point of .

Proof. Let for
Let, if possible, no point of be a limit point of . Then, for each , there exists open set such that and . Since is soft base, there exists such that Therefore, . Moreover, if and be any two points such that which means either or , then there exists and in such that and . Now, which guarantees that which implies that which implies . Thus, there exists a one to one correspondence of on to Now, being uncountable, it follows that is uncountable. This is a contradiction.

Theorem 9. Let and be two and suppose be a continuous function such that is continuous function and let supposes the Then, safely, has the

Proof. Suppose be an infinite subset of , so that contains an enumerable set , then there exists enumerable set such that has implying that every infinite soft subset supposes accumulation point belonging to , and this implies that has limit point, say, implies that the limit of NSQ sequence is is continuous. Furthermore, implies that limit of a sequence is implying that limit of a sequence . Finally, we have shown that there exists infinite soft subset of containing a limit point ; this guarantees that has

Theorem 10. Let and let be a sequence in such that it converges to a point , then the set consisting of points and is soft compact.

Proof. Given and let be an sequence in such that it converges to a point , that is Let . Let be open covering of so that implies that there exists such that By convergence, implies that there exists and . Evidently, contains the points Look carefully at the points and train them in a way as generating a finite soft set. Let Then, For this Hence, there exists such that . Evidently, . This shows that is open cover of Thus, an arbitrary NS open cover of is reducible to a finite subcover , and it follows that is compact.

Theorem 11. If such that it has the characteristics of NSQ sequentially compactness. Then, is countably compact.

Proof. Let and let be finite soft subset of . Let be an sequence of points of Then, being finite, at least one of the elements in say must be duplicated an infinite number of times in the NSQ sequence. Hence, is soft NSQ subsequence of such that it is constant sequence and repeatedly constructed by single soft number , and we know that a soft constant sequence converges on its self. So, it converges to which belongs to Hence, is soft sequentially compact.

Theorem 12. Let and be another . Let a soft continuous mapping of an sequentially compact space into , then is sequentially compact.

Proof. Given and be another . Let be a continuous mapping of a sequentially compact space into , then we have to prove sequentially. For this, we proceed as follows. Let be a sequence of points. Then, for each , there exists . Thus, we obtain an sequence in . But being soft sequentially compact, there is an subsequence of
such that So, by continuity of implies that Thus, is a soft subsequence of converges to in Hence, sequentially compact.

Theorem 13. Let and suppose be two continuous function on an NSQTS in to an NSQTS which is Hausdorff. Then, soft set is closed of

Proof. Let if is an set of function. If , it is clearly open, and therefore, is closed, that is, nothing is proved in this case. Let us consider the case when and let Then, does not belong . Result in Now, being Hausdorff space, so there exists open sets of and , respectively, such that and such that these sets such that the possibility of one rules out the possibility of other. By soft continuity of as well as is open nhd of , and therefore, is contained in for, and because and are mutually exclusive. This implies that does not belong to Therefore, this shows that is nhd of each of its points. So, open, and hence is closed.

Theorem 14. Let such that it is Hausdorff space and let be soft continuous function of into itself. Then, the NSQ set of fixed points under is a closed set.

Proof. Let . If Then, is open, and therefore, closed. So, let and let .Then, does not belong to , and therefore, . Now, , being two distinct points of the Hausdorff space so there exist open sets such that and are disjoint. Also, by the continuity, is an open set containing we pretend that . Since . As implies that μ does not belong to .Therefore, . Thus, is the neighborhood of each of its points. So, is open, and hence, is closed.

6. Conclusion

Neutrosophic soft topology (NST) is extension of vague soft topology (VST) and VST is extension of fuzzy soft topology (FST). VST gives two type of informations. One is true, and second one is false. It does not give informations about the indeterminacy (doubtful) case. NST is dominant over VST because it supposes all the three informations that is true, false, and indeterminacy at the same time. NST has narrow domain as compared to neutrosophic soft bitopology (NSBT). Some problems are very hard to discuss in NST, so need of NSBT is felt. Still NSTS is unable to afford problems of large number of domain. So, extension is needed to neutrosophic soft tritopology (NSTT) and neutrosophic soft quad-topology (NSQT). In our work, we regenerated some structures in NSQTS with new definition that is open sets relative to soft points. We worked with the operations given in references [14–16] which are entirely different from references [13, 17]. In future, we will develop neutrosophic soft penta-topology and neutrosophic soft hexa-topology relative to soft points of the space under more generalized neutrosophic soft open sets. We will try to develop neutrosophic soft separation axioms and their engagement with each other. In addition to this, neutrosophic soft other separation axioms will also be given special attention. After doing this, on the basis of reference [12], we will try to build the same structures with respect to hyper-soft sets and then with respect to plithogenic hyper-soft sets.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest to report regarding the present study.

Authors’ Contributions

All authors read and approved the final manuscript.

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Copyright © 2022 Arif Mehmood et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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